A Tight Linear Time (1/2)-Approximation for Unconstrained

A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular
Maximization
Niv Buchbinder†
Moran Feldman
Joseph (Seffi) Naor∗
Roy Schwartz
Statistics and Operations Research Dept. Computer Science Dept. Computer Science Dept.
Theory Group
Tel Aviv University
Technion
Technion
Microsoft Research
Tel Aviv, Israel
Haifa, Israel
Haifa, Israel
Redmond, WA
[email protected]
[email protected] [email protected] [email protected]
Abstract—We consider the Unconstrained Submodular
Maximization problem in which we are given a non-negative
submodular function f : 2N → R+ , and the objective is to
find a subset S ⊆ N maximizing f (S). This is one of the
most basic submodular optimization problems, having a wide
range of applications. Some well known problems captured
by Unconstrained Submodular Maximization include MaxCut, Max-DiCut, and variants of Max-SAT and maximum
facility location. We present a simple randomized linear time
algorithm achieving a tight approximation guarantee of 1/2,
thus matching the known hardness result of Feige et al.
[11]. Our algorithm is based on an adaptation of the greedy
approach which exploits certain symmetry properties of the
problem. Our method might seem counterintuitive, since it is
known that the greedy algorithm fails to achieve any bounded
approximation factor for the problem.
Keywords-Submodular
rithms
Functions,
Approximation
Algo-
I. I NTRODUCTION
The study of combinatorial problems with submodular
objective functions has recently attracted much attention, and
is motivated by the principle of economy of scale, prevalent
in real world applications. Submodular functions are also
commonly used as utility functions in economics and algorithmic game theory. Submodular functions and submodular
maximization play a major role in combinatorial optimization, where several well known examples of submodular
functions include cuts in graphs and hypergraphs, rank
functions of matroids, and covering functions. A function
f : 2N → R+ is called submodular if for every A ⊆ B ⊆ N
and u ∈ N , f (A ∪ {u}) − f (A) ≥ f (B ∪ {u}) − f (B)1 .
Submodularity captures the principle of diminishing returns
in economics.
Perhaps the most basic submodular maximization problem is Unconstrained Submodular Maximization (USM).
Given a non-negative submodular fucntion f , the goal is to
† Research supported in part by ISF grant 954/11, and by BSF grant
2010426.
∗ Research supported in part by the Google Inter-university center for
Electronic Markets, by ISF grant 954/11, and by BSF grant 2010426.
1 An equivalent definition is that for every A, B ⊆ N : f (A) + f (B) ≥
f (A ∪ B) + f (A ∩ B).
find a subset S ⊆ N maximizing f (S). Note that there is
no restriction on the choice of S, as any subset of N is a
feasible solution. USM captures many well studied problems
such as Max-Cut, Max-DiCut [15], [20], [22], [25], [27],
[36], variants of Max-SAT, and maximum facility location
[1], [7], [8]. Moreover, USM has various applications in
more practical settings such as marketing in social networks
[21], revenue maximization with discrete choice [2], and
algorithmic game theory [10], [34].
USM has been studied starting since the sixties in the
operations research community [2], [6], [16], [17], [18], [26],
[29], [33]. Not surprisingly, as USM captures NP-hard problems, all these works provide algorithms that either solve
special cases of the problem, or provide exact algorithms
whose time complexity cannot be efficiently bounded, or
provide efficient algorithms whose output has no provable
guarantee.
The first rigorous study of the USM problem was conducted by Feige et al. [11], who provided several constant
approximation factor algorithms. They proved that a subset
S chosen uniformly at random is a (1/4)-approximation.
Additionally, they also described two local search algorithms. The first uses f as the objective function, and provides an approximation of 1/3. The second uses a noisy version of f as the objective function, and achieves an improved
approximation guarantee of 2/5. Gharan and Vondr´ak [14]
showed that an extension of the last method, known as simulated annealing, can provide an improved approximation of
roughly 0.41. Their algorithm, like that of Feige et al. [11],
uses local search with a noisy objective function. However,
in [14] the noise decreases as the algorithm advances, as
opposed to being constant as in [11]. Feldman et al. [12]
observed that if the simulated annealing algorithm of [14]
outputs a relatively poor solution, then it must generate at
some point a set S which is structurally similar to some
optimal solution. Moreover, they showed that this structural
similarity can be traded for value, providing an overall
improved approximation of roughly 0.42.
It is important to note that for many special cases of USM,
better approximation factors are known. For example, the
seminal work of Goemans and Williamson [15] provides an
0.878-approximation for Max-Cut based on a semidefinite
programming formulation, and Ageev and Sviridenko [1]
provide an approximation of 0.828 for the maximum facility
location problem.
On the negative side, Feige et al. [11] studied the hardness
of USM assuming the function f is given via a value
oracle2 . They proved that for any constant ε > 0, any
algorithm achieving an approximation of (1/2 + ε) requires
an exponential number of oracle queries. This hardness
result holds even if f is symmetric, in which case it is
known to be tight [11]. Recently, Dobzinski and Vondr´ak [9]
proved that even if f has a compact representation (which
is part of the input), the latter hardness still holds assuming
RP ̸= N P .
A. Our Results
In this paper we resolve the approximability of the Unconstrained Submodular Maximization problem. We design a tight linear time randomized (1/2)-approximation for
the problem. We begin by presenting a simple greedy-based
deterministic algorithm that achieves a (1/3)-approximation
for USM.
Theorem I.1. There exists a deterministic linear time (1/3)approximation algorithm for the Unconstrained Submodular Maximization problem.
We show that our analysis of the algorithm is tight by
providing an instance for which the algorithm achieves an
approximation of 1/3 + ε for an arbitrary small ε > 0.
The improvement in the performance is achieved by incorporating randomness into the choices of the algorithm. We
obtain a tight bound for USM without increasing the time
complexity of the algorithm.
Theorem I.2. There exists a randomized linear time (1/2)approximation algorithm for the Unconstrained Submodular Maximization problem.
In both Theorems I.1 and I.2 we assume that a single
query to the value oracle takes O(1) time. Thus, a linear
time algorithm is an algorithm which makes O(n) oracle
queries plus O(n) other operations, where n is the size of
the ground set N .
Building on the above two theorems, we provide two
additional approximation algorithms for Submodular MaxSAT and Submodular Welfare with 2 players (for the exact
definition of these problems please refer to Section IV). Both
problems are already known to have a tight approximation
[13]. However, our algorithms run in linear time, thus,
2 The
explicit representation of a submodular function might be exponential in the size of its ground set. Thus, it is standard to assume that
the function is accessed via a value oracle. For a submodular function
f : 2N → R+ , given a set S ⊆ N , the value oracle returns the value of
f (S).
significantly improving the time complexity, while achieving
the same performance guarantee.
Theorem I.3. There exists a randomized linear time (3/4)approximation algorithm for the Submodular Max-SAT
problem.
Theorem I.4. There exists a randomized linear time (3/4)approximation algorithm for the Submodular Welfare
problem with 2 players .
B. Techniques
The algorithms we present are based on a greedy approach. It is known that a straightforward greedy approach
fails for USM. In contrast, Feldman et al. [13] recently
showed that a continuous greedy approach does provide
a (1/e)-approximation for maximizing any non-monotone
submodular function over a matroid. Recall that better
bounds than 1/e are already known for USM.
To understand the main ideas of our algorithm, consider
a non-negative submodular function f . Let us examine
the complement of f , denoted by f¯, defined as f¯(S) ,
f (N \ S), for any S ⊆ N . Note that since f is submodular,
f¯ is also submodular. Additionally, given an optimal solution
OP T ⊆ N for USM with input f , N \ OP T is an
optimal solution for f¯, and both solutions have the exact
same value. Consider, for example, the greedy algorithm.
For f , it starts from an empty solution and iteratively adds
elements to it in a greedy fashion. However, when applying
the greedy algorithm to f¯, one gets an algorithm for f that
effectively starts with the solution N and iteratively removes
elements from it. Both algorithms are equally reasonable,
but, unfortunately, both fail.
Despite the failure of the greedy algorithm when applied
separately to either f or f¯, we show that a correlated
execution on both f and f¯ provides a much better result.
That is, we start with two solutions ∅ and N . The algorithm
considers the elements (in arbitrary order) one at a time.
For each element it determines whether it should be added
to the first solution or removed from the second solution.
Thus, after a single pass over the ground set N , both
solutions completely coincide. This is the solution that the
algorithm outputs. We show that a greedy choice in each
step obtains an approximation guarantee of 1/3, hence
proving Theorem I.1. To get Theorem I.2, we use a natural
extension of the greedy rule, together with randomization,
thus improving the approximation guarantee to 1/2.
C. Related Work
Many works consider the problem of maximizing a nonnegative submodular function subject to various combinatorial constraints defining the feasible solutions [13], [19],
[30], [31], [39], or minimizing a submodular function subject
to such constraints [23], [24], [35]. Interestingly, the problem
of unconstrained submodular minimization can be solved in
polynomial time [32].
Another line of work deals with maximizing normalized
monotone submodular functions, again, subject to various
combinatorial constraints. A continuous greedy algorithm
was given by Calinescu et al. [3] for maximizing a normalized monotone submodular function subject to a matroid
constraint. Later, Lee et al. [31] gave a local search algorithm achieving 1/p − ε approximation for maximizing such
functions subject to the intersection of p matroids. Kulik
et al. [28] showed a 1 − 1/e − ε approximation algorithm
for maximizing a normalized monotone submodular function
subject to multiple knapsack constraints. Recently, Chekuri
et al. [5] and Feldman et al. [13] gave non-monotone
counterparts of the continuous greedy algorithm of [3], [38].
These results improve several non-monotone submodular
optimization problems. Some of the above results were
generalized by Chekuri et al. [4], who provide a dependent
rounding technique for various polytopes, including matroid
and matroid-intersection polytops. The advantage of this
rounding technique is that it guarantees strong concentration
bounds for submodular functions. Additionally, Chekuri et
al. [5] define a contention resolution rounding scheme which
allows one to obtain approximations for different combinations of constraints.
II. A D ETERMINISTIC (1/3)-A PPROXIMATION
A LGORITHM
In this section we present a deterministic linear time
algorithm for USM. The algorithm proceeds in n iterations
that correspond to some arbitrary order u1 , . . . , un of the
ground set N . The algorithm maintains two solutions X and
Y . Initially, we set the solutions to X0 = ∅ and Y0 = N .
In the ith iteration the algorithm either adds ui to Xi−1
or removes ui from Yi−1 . This decision is done greedily
based on the marginal gain of each of the two options.
Eventually, after n iterations both solutions coincide, and
we get Xn = Yn ; this is the output of the algorithm. A
formal description of the algorithm appears as Algorithm 1.
Algorithm 1: DeterministicUSM(f, N )
6
X0 ← ∅, Y0 ← N .
for i = 1 to n do
ai ← f (Xi−1 ∪ {ui }) − f (Xi−1 ).
bi ← f (Yi−1 \ {ui }) − f (Yi−1 ).
if ai ≥ bi then Xi ← Xi−1 ∪ {ui }, Yi ← Yi−1 .
else Xi ← Xi−1 , Yi ← Yi−1 \ {ui }.
7
return Xn (or equivalently Yn ).
1
2
3
4
5
The rest of this section is devoted for proving Theorem I.1, i.e., we prove that the approximation guarantee
of Algorithm 1 is 1/3. Denote by ai the change in value
of the first solution if element ui is added to it in the
ith iteration, i.e., f (Xi−1 ∪ {ui }) − f (Xi−1 ). Similarly,
denote by bi the change in value of the second solution
if element ui is removed from it in the ith iteration, i.e.,
f (Yi−1 \{ui })−f (Yi−1 ). We start with the following useful
lemma.
Lemma II.1. For every 1 ≤ i ≤ n, ai + bi ≥ 0.
Proof: Notice that (Xi−1 ∪ {ui }) ∪ (Yi \ {ui }) = Yi−1
and (Xi−1 ∪{ui })∩(Yi \{ui }) = Xi−1 . By combining both
observations with submodularity, one gets:
ai + bi = [f (Xi−1 ∪ {ui }) − f (Xi−1 )] +
[f (Yi−1 \ {ui }) − f (Yi−1 )]
= [f (Xi−1 ∪ {ui }) + f (Yi−1 \ {ui })] −
[f (Xi−1 ) + f (Yi−1 )] ≥ 0 .
Let OP T denote an optimal solution. Define OP Ti ,
(OP T ∪ Xi ) ∩ Yi . Thus, OP Ti coincides with Xi and Yi on
elements 1, . . . , i, and it coincides with OP T on elements
i + 1, . . . , n. Note that OP T0 = OP T and the output of
the algorithm is OP Tn = Xn = Yn . Examine the sequence
f (OP T0 ), . . . , f (OP Tn ), which starts with f (OP T ) and
ends with the value of the output of the algorithm. The
main idea of the proof is to bound the total loss of value
along this sequence. This goal is achieved by the following
lemma which upper bounds the loss in value between every
two consecutive steps in the sequence. Formally, the loss
of value, i.e., f (OP Ti−1 ) − f (OP Ti ), is no more than the
total increase in value of both solutions maintained by the
algorithm, i.e., [f (Xi ) − f (Xi−1 )] + [f (Yi ) − f (Yi−1 )].
Lemma II.2. For every 1 ≤ i ≤ n,
f (OP Ti−1 )−f (OP Ti ) ≤
[f (Xi ) − f (Xi−1 )] + [f (Yi ) − f (Yi−1 )] .
Before proving Lemma II.2, let us show that Theorem I.1
follows from it.
Proof of Theorem I.1: Summing up Lemma II.2 for
every 1 ≤ i ≤ n gives:
n
∑
i=1
n
∑
i=1
[f (OP Ti−1 ) − f (OP Ti )] ≤
[f (Xi ) − f (Xi−1 )] +
n
∑
[f (Yi ) − f (Yi−1 )] .
i=1
The above sum is telescopic. Collapsing it, we get:
f (OP T0 ) − f (OP Tn ) ≤
[f (Xn ) − f (X0 )] + [f (Yn ) − f (Y0 )] ≤ f (Xn ) + f (Yn ) .
Recalling the definitions of OP T0 and OP Tn , we obtain
that f (Xn ) = f (Yn ) ≥ f (OP T )/3.
It is left to prove Lemma II.2.
Figure 1. Tight example for Algorithm 1. The weight of the dashed edges
is 1 − ε. All other edges have weight of 1.
Proof of Lemma II.2: Assume without loss of generality that ai ≥ bi , i.e., Xi ← Xi−1 ∪ {ui } and Yi ← Yi−1
(the other case is analogous). Notice that in this case
OP Ti = (OP T ∪Xi )∩Yi = OP Ti−1 ∪{ui } and Yi = Yi−1 .
Thus the inequality that we need to prove becomes:
f (OP Ti−1 )−f (OP Ti−1 ∪{ui }) ≤ f (Xi )−f (Xi−1 ) = ai .
We now consider two cases. If ui ∈ OP T , then the left
hand side of the last inequality is 0, and all we need to show
is that ai is non-negative. This is true since ai + bi ≥ 0 by
Lemma II.1, and ai ≥ bi by our assumption.
If ui ̸∈ OP T , then also ui ̸∈ OP Ti−1 , and thus:
f (OP Ti−1 )−f (OP Ti−1 ∪ {ui }) ≤
f (Yi−1 \ {ui }) − f (Yi−1 ) = bi ≤ ai .
The first inequality follows by submodularity: OP Ti−1 =
((OP T ∪Xi−1 )∩Yi−1 ) ⊆ Yi−1 \{ui } (recall that ui ∈ Yi−1
and ui ̸∈ OP Ti−1 ). The second inequality follows from our
assumption that ai ≥ bi .
A. Tight Example
We now show that the analysis of Algorithm 1 is tight.
Theorem II.3. For an arbitrarily small constant ε > 0,
there exists a submodular function for which Algorithm 1
provides only (1/3 + ε)-approximation.
Proof: The proof follows by analyzing Algorithm 1 on
the cut function of the weighted digraph given in Figure 1.
The maximum weight cut in this digraph is {u1 , u4 , u5 }.
This cut has weight of 6 − 2ε. We claim that Algorithm
1 outputs the cut {u2 , u3 , u4 , u5 }, whose value is only 2
(assuming the nodes of the graph are considered at the given
order). Hence, the approximation guarantee of Algorithm 1
on the above instance is:
1
1
2
=
≤ +ε .
6 − 2ε
3−ε
3
Let us track the execution of the algorithm. Let Xi , Yi be
the solutions maintained by the algorithm. Initially X0 =
∅, Y0 = {u1 , u2 , u3 , u4 , u5 }. Note that in case of a tie (ai =
bi ), Algorithm 1 takes node ui .
1) In the first iteration the algorithm considers u1 . Adding
this node to X0 increases the value of this solution
by 2 − 2ε. On the other hand, removing this node
from Y0 increases the value of Y0 by 2. Hence, X1 ←
X0 , Y1 ← Y0 \ {u1 }.
2) Let us inspect the next two iterations in which the
algorithm considers u2 , u3 . One can easily verify
that these two iterations are independent, hence, we
consider only u2 . Adding u2 to X1 increases its
value by 1. On the other hand, removing u2 from
Y1 = {u2 , u3 , u4 , u5 } also increases the value of Y1
by 1. Thus, the algorithm adds u2 to X1 . Since u2
and u3 are symmetric, the algorithm also adds u3 to
X1 . Thus, at the end of these two iterations X3 =
X1 ∪ {u2 , u3 } = {u2 , u3 }, Y3 = {u2 , u3 , u4 , u5 }.
3) Finally, the algorithm considers u4 and u5 . These two
iterations are also independent so we consider only
u4 . Adding u4 to X3 does not increase the value of
X3 . Also removing u4 from Y3 does not increase the
value of Y3 . Thus, the algorithm adds u4 to X3 . Since
u4 and u5 are symmetric, the algorithm also adds u5
to X3 . Thus, we get X5 = Y5 = {u2 , u3 , u4 , u5 }.
III. A R ANDOMIZED (1/2)-A PPROXIMATION
A LGORITHM
In this section we present a randomized algorithm achieving a tight (1/2)-approximation for USM. Algorithm 1
compared the marginal profits ai and bi , and based on
this comparison it made a greedy deterministic decision
whether to include or exclude ui from its output. The random
algorithm we next present makes a “smoother” decision,
based on the values ai and bi . In each step, it randomly
chooses whether to include or exclude ui from the output
with probability derived from the values ai and bi . A formal
description of the algorithm appears as Algorithm 2.
The rest of this section is devoted to proving that Algorithm 2 provides an approximation guarantee of 1/2 to
USM. In Appendix A, we describe another algorithm which
can be thought of as the continuous counterpart of Algorithm 2. This algorithm achieves the same approximation
ratio (up to low order terms), but keeps a fractional inner
state.
Let us first introduce some notation. Notice that for
every 1 ≤ i ≤ n, Xi and Yi are random variables
denoting the sets of elements in the two solutions generated
by the algorithm at the end of the ith iteration. As in
Section II, let us define the following random variables:
OP Ti , (OP T ∪ Xi ) ∩ Yi . Note that as before, X0 = ∅,
Y0 = N and OP T0 = OP T . Moreover, the following
Algorithm 2: RandomizedUSM(f, N )
1
2
3
4
5
6
7
8
X0 ← ∅, Y0 ← N .
for i = 1 to n do
ai ← f (Xi−1 ∪ {ui }) − f (Xi−1 ).
bi ← f (Yi−1 \ {ui }) − f (Yi−1 ).
a′i ← max{ai , 0}, b′i ← max{bi , 0}.
with probability a′i /(a′i + b′i )* do:
Xi ← Xi−1 ∪ {ui }, Yi ← Yi−1 .
else (with the compliment probability b′i /(a′i + b′i ))
do: Xi ← Xi−1 , Yi ← Yi−1 \ {ui }.
return Xn (or equivalently Yn ).
*
If a′i = b′i = 0, we assume a′i /(a′i + b′i ) = 1.
always holds: OP Tn = Xn = Yn . The proof idea is similar
to that of the deterministic algorithm in Section II when
considering the sequence E[f (OP T0 )], . . . , E[f (OP Tn )].
This sequence starts with f (OP T ) and ends with the
expected value of the algorithm’s output. The following
lemma upper bounds the loss of two consecutive elements
in the sequence. Formally, E[f (OP Ti−1 ) − f (OP Ti )] is
upper bounded by the average expected change in the
value of the two solutions maintained by the algorithm, i.e.,
1/2 · E [(f (Xi ) − f (Xi−1 ) + f (Yi ) − f (Yi−1 )].
Lemma III.1. For every 1 ≤ i ≤ n,
E [f (OP Ti−1 ) − f (OP Ti )] ≤
1
· E [(f (Xi ) − f (Xi−1 ) + f (Yi ) − f (Yi−1 )] , (1)
2
where expectations are taken over the random choices of the
algorithm.
Before proving Lemma III.1, let us show that Theorem I.2
follows from it.
Proof of Theorem I.2: Summing up Lemma III.1 for
every 1 ≤ i ≤ n yields:
n
∑
E [f (OP Ti−1 ) − f (OP Ti )] ≤
i=1
n
1 ∑
·
E [f (Xi ) − F (Xi−1 ) + f (Yi ) − F (Yi−1 )] .
2 i=1
The above sum is telescopic. Collapsing it, we get:
E [f (OP T0 ) − f (OP Tn )] ≤
1
· E [f (Xn ) − f (X0 ) + f (Yn ) − f (Y0 )] ≤
2
E[f (Xn ) + f (Yn )]
.
2
Recalling the definitions of OP T0 and OP Tn , we obtain
that E[f (Xn )] = E[f (Yn )] ≥ f (OP T )/2.
It is left to prove Lemma III.1.
Proof of Lemma III.1: Notice that it suffices to prove
Inequality (1) conditioned on any event of the form Xi−1 =
Si−1 , when Si−1 ⊆ {u1 , . . . , ui−1 } and the probability
that Xi−1 = Si−1 is non-zero. Hence, fix such an event
for a given Si−1 . The rest of the proof implicitly assumes
everything is conditioned on this event. Notice that due to
the conditioning, the following quantities become constants:
• Yi−1 = Si−1 ∪ {ui , . . . , un }.
• OP Ti−1 , (OP T ∪ Xi−1 ) ∩ Yi−1 =
Si−1 ∪ (OP T ∩ {ui , . . . , un }).
• ai and bi .
Moreover, by Lemma II.1, ai + bi ≥ 0. Thus, it cannot be
that both ai , bi are strictly less than zero. Hence, we only
need to consider the following 3 cases:
Case 1 (ai ≥ 0 and bi ≤ 0): In this case a′i /(a′i +
′
bi ) = 1, and so the following always happen: Yi = Yi−1 =
Si−1 ∪ {ui , . . . , un } and Xi ← Si−1 ∪ {ui }. Hence, f (Yi ) −
f (Yi−1 ) = 0. Also, by our definition OP Ti = (OP T ∪Xi )∩
Yi = OP Ti−1 ∪ {ui }. Thus, we are left to prove that:
f (OP Ti−1 ) − f (OP Ti−1 ∪ {ui }) ≤
1
ai
· [f (Xi ) − f (Xi−1 )] =
.
2
2
If ui ∈ OP T , then the left hand side of the last expression
is 0, which is clearly no larger than the non-negative ai /2.
If ui ∈
̸ OP T , then:
f (OP Ti−1 ) − f (OP Ti−1 ∪ {ui }) ≤
f (Yi−1 \ {ui }) − f (Yi−1 ) = bi ≤ 0 ≤ ai /2 .
The first inequality follows from submodularity since
OP Ti−1 , (OP T ∪ Xi−1 ) ∩ Yi−1 ⊆ Yi−1 \ {ui } (note
that ui ∈ Yi−1 and ui ̸∈ OP Ti−1 ).
Case 2 (ai < 0 and bi ≥ 0): This case is analogous
to the previous one, and therefore, we omit its proof.
Case 3 (ai ≥ 0 and bi > 0): In this case a′i = ai , b′i =
bi . Therefore, with probability ai /(ai + bi ) the following
events happen: Xi ← Xi−1 ∪ {ui } and Yi ← Yi−1 , and
with probability bi /(ai + bi ) the following events happen:
Xi ← Xi−1 and Yi ← Yi−1 \ {ui }. Thus,
E[f (Xi ) − f (Xi−1 ) + f (Yi ) − f (Yi−1 )] =
ai
· [f (Xi−1 ∪ {ui }) − f (Xi−1 )] +
ai + bi
bi
· [f (Yi−1 \ {ui }) − f (Yi−1 )]
ai + bi
a2 + b2i
= i
.
ai + bi
(2)
Next, we upper bound E[f (OP Ti−1 ) − f (OP Ti )]. As
OP Ti = (OP T ∪ Xi ) ∩ Yi , we get,
E[f (OP Ti−1 ) − f (OP Ti )] =
ai
· [f (OP Ti−1 ) − f (OP Ti−1 ∪ {ui })] +
ai + bi
bi
· [f (OP Ti−1 ) − f (OP Ti−1 \ {ui })]
ai + bi
ai bi
≤
.
(3)
ai + bi
The final inequality follows by considering two cases. Note
first that ui ∈ Yi−1 and ui ̸∈ Xi−1 . If ui ̸∈ OP Ti−1 then
the second term of the left hand side of the last inequality
equals zero. Moreover, OP Ti−1 , (OP T ∪ Xi−1 ) ∩ Yi−1 ⊆
Yi−1 \ {ui }, and therefore, by submodularity,
f (OP Ti−1 )−f (OP Ti−1 ∪ {ui }) ≤
f (Yi−1 \ {ui }) − f (Yi−1 ) = bi .
If ui ∈ OP Ti−1 then the first term of the left hand side
of the inequality equals zero, and we also have Xi−1 ⊆
((OP T ∪ Xi−1 ) ∩ Yi−1 ) \ {ui } = OP Ti−1 \ {ui }. Thus, by
submodularity,
f (OP Ti−1 )−f (OP Ti−1 \ {ui }) ≤
f (Xi−1 ∪ {ui }) − f (Xi−1 ) = ai .
Plugging (2) and (3) into Inequality (1), we get the following:
( 2
)
ai bi
1
ai + b2i
≤ ·
,
ai + bi
2
ai + bi
which can be easily verified.
IV. A PPROXIMATION A LGORITHMS FOR S UBMODULAR
M AX -SAT AND S UBMODULAR W ELFARE WITH 2
P LAYERS
In this section we build upon ideas from the previous
sections to obtain linear time tight (3/4)-approximation
algorithms for both the Submodular Max-SAT (SSAT) and
Submodular Welfare (SW) with 2 players problems. In
contrast to USM, tight approximations are already known for
the above two problems [13]. However, the algorithms we
present in this work, in addition to providing tight approximations, also run in linear time. We require the definition
of a monotone submodular function: a submodular function
f : 2N → R+ is monotone if for every A ⊆ B ⊆ N ,
f (A) ≤ f (B). Moreover, a monotone submodular function
f is normalized if f (∅) = 0.
A. A (3/4)-Approximation for Submodular Max-SAT
In the Submodular Max-SAT problem we are given a
CNF formula Ψ with a set C of clauses over a set N of
variables, and a normalized monotone submodular function
f : 2C → R+ over the set of clauses. Given an assignment
ϕ : N → {0, 1}, let C(ϕ) ⊆ C be the set of clauses satisfied
by ϕ. The goal is to find an assignment ϕ that maximizes
f (C(ϕ)).
Usually, an assignment ϕ can give each variable exactly a
single truth value. However, for the sake of the algorithm we
extend the notion of assignments, and think of an extended
assignment ϕ′ which is a relation ϕ′ ⊆ N × {0, 1}. That is,
the assignment ϕ′ can assign up to 2 truth values to each
variable. A clause C is satisfied by an (extended) assignment
ϕ′ , if there exists a positive literal in the clause which is
assigned to truth value 1, or there exists a negative literal
in the clause which is assigned a truth value 0. Note again
that it might happen that some variable is assigned both 0
and 1. Note also, that an assignment is a feasible solution
to the original problem if and only if it assigns exactly one
truth value to every variable of N . Let C(ϕ′ ) be the set of
clauses satisfied by ϕ′ . We define g : N × {0, 1} → R+
using g(ϕ′ ) , f (C(ϕ′ )). Using this notation, we can restate
SSAT as the problem of maximizing the function g over the
set of feasible assignments.
Observation IV.1. The function g is a normalized monotone
submodular function.
The algorithm we design for SSAT conducts n iterations.
It maintains at each iteration 1 ≤ i ≤ n two assignments Xi
and Yi which always satisfy that: Xi ⊆ Yi . Initially, X0 = ∅
assigns no truth values to the variables, and Y0 = N ×{0, 1}
assigns both truth values to all variables. The algorithm
considers the variables in an arbitrary order u1 , u2 , . . . , un .
For every variable ui , the algorithm evaluates the marginal
profit from assigning it only the truth value 0 in both
assignments, and assigning it only the truth value 1 in both
assignments. Based on these marginal values, the algorithm
makes a random decision on the truth value assigned to ui .
After the algorithm considers a variable ui , both assignments
Xi and Yi agree on the single truth value assigned to ui .
Thus, when the algorithm terminates both assignments are
identical and feasible. A formal statement of the algorithm
appears as Algorithm 3.
The proof of the Theorem I.3 uses similar ideas to
previous proofs in this paper, and is therefore, deferred to
Appendix B. Note, however, that unlike for the previous
algorithms, here, the fact that the algorithm runs in linear
time requires a proof. The source of the difficulty is that
we have an oracle access to f , but use the function g in
the algorithm. Therefore, we need to prove that we may
implement all oracle queries of g that are conducted during
the execution of the algorithm in linear time.
A Note on Max-SAT: The well known Max-SAT
problem is in fact a special case of SSAT where f is a
linear function. We note that Algorithm 3 can be applied
to Max-SAT in order to achieve a (3/4)-approximation in
linear time, however, this is not immediate. This result is
summarized in the following theorem. The proof of this
theorem is deferred to a full version of the paper.
Algorithm 3: RandomizedSSAT(f, Ψ)
11
X0 ← ∅, Y0 ← N × {0, 1}.
for i = 1 to n do
ai,0 ← g(Xi−1 ∪ {ui , 0}) − g(Xi−1 ).
ai,1 ← g(Xi−1 ∪ {ui , 1}) − g(Xi−1 ).
bi,0 ← g(Yi−1 \ {ui , 0}) − g(Yi−1 ).
bi,1 ← g(Yi−1 \ {ui , 1}) − g(Yi−1 ).
si,0 ← max{ai,0 + bi,1 , 0}.
si,1 ← max{ai,1 + bi,0 , 0}.
with probability si,0 /(si,0 + si,1 )* do:
Xi ← Xi−1 ∪ {ui , 0}, Yi ← Yi−1 \ {ui , 1}.
else (with the compliment probability
si,1 /(si,0 + si,1 )) do:
Xi ← Xi−1 ∪ {ui , 1}, Yi ← Yi−1 \ {ui , 0}.
12
return Xn (or equivalently Yn ).
1
2
3
4
5
6
7
8
9
10
*
If si,0 = si,1 = 0, we assume si,0 /(si,0 + si,1 ) = 1.
Theorem IV.2. Algorithm 3 has a linear time implementation for instances of Max-SAT.
B. A (3/4)-Approximation for Submodular Welfare with 2
Players
The input for the Submodular Welfare problem consists
of a ground set N of n elements and k players, each
equipped with a normalized monotone submodular utility
function fi : 2N → R+ . The goal is to divide the elements
among the players while maximizing the social welfare. Formally, the objective∑is to partition N into N1 , N2 , . . . , Nk
k
while maximizing i=1 fi (Ni ).
We give below two different short proofs of Theorem I.4
via reductions to SSAT and USM, respectively. The second
proof is due to Vondr´ak [37].
Proof of Theorem I.4: We provide here two proofs.
Proof (1): Given an instance of SW with 2 players,
construct an instance of SSAT as follows:
1) The set of variables is N .
2) The CNF formula Ψ consists of 2|N | singleton
clauses; one for every possible literal.
3) The objective function f : 2C → R+ is defined as
following. Let P ⊆ C be the set of clauses of Ψ
consisting of positive literals. Then, f (C) = f1 (C ∩
P ) + f2 (C ∩ (C \ P )).
Every assignment ϕ to this instance of SSAT corresponds
to a solution of SW using the following rule: N1 = {u ∈
N |ϕ(u) = 0} and N2 = {u ∈ N |ϕ(u) = 1}. One can
easily observe that this correspondence is reversible, and
that each assignment has the same value as the solution
it corresponds to. Hence, the above reduction preserves
approximation ratios.
Moreover, queries of f can be answered in constant time
using the following technique. We track for every subset
C ⊆ C in the algorithm the subsets C ∩ P and C ∩ (C \ P ).
For Algorithm 3 this can be done without effecting its
running time. Then, whenever the value of f (C) is queried,
answering it simply requires making two oracle queries:
f1 (C ∩ P ) and f2 (C ∩ (C \ P )).
Proof (2): In any feasible solution to SW with two
players, the set N1 uniquely determines the set N2 = N −
N1 . Hence, the value of the solution as a function of N1 is
given by g(N1 ) = f1 (N1 ) + f2 (N − N1 ). Thus, SW with
two players can be restated as the problem of maximizing
the function g over the subsets of N .
Observe that the function g is a submodular function, but
unlike f1 and f2 , it is possibly non-monotone. Moreover,
we can answer queries to the function g using only two
oracle queries to f1 and f2 3 . Thus, we obtain an instance
of USM. We apply Algorithm 2 to this instance. Using
the analysis of Algorithm 2 as is, provides only a (1/2)approximation for our problem. However, by noticing that
g(∅) + g(N ) ≥ f1 (N ) + f2 (N ) ≥ g(OP T ), the claimed
(3/4)-approximation is obtained.
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A PPENDIX
A. A (1/2)-Approximation for USM using Fractional Values
In this section we present Algorithm 4, which is the continuous counterpart of Algorithm 2, presented in Section III.
In order to describe Algorithm 4, we need some notation. Given a submodular function f : N → R+ , its
multilinear extension is a function F : [0, 1]N → R+ ,
whose value at point x ∈ [0, 1]N is the expected value
of f over a random subset R(x) ⊆ N . The subset
R(x) contains each element u ∈ N independently with
N
probability x
u . Formally,∏for every
∑
∏ x ∈ [0, 1] , F (x) ,
E[R(x)] = S⊆N f (S) u∈S xu u∈S
/ (1 − xu ). For two
vectors x, y ∈ [0, 1]N , we use x ∨ y and x ∧ y to denote
the coordinate-wise maximum and minimum, respectively,
of x and y (formally, (x ∨ y)u = max{xu , yu } and
(x ∧ y)u = min{xu , yu }).
We abuse notations both in the description of the algorithm and in its analysis, and unify a set with its characteristic vector. We also assume that we have an oracle access
to F . If this is not the case, then one can approximate the
value of F arbitrarily well using sampling. This makes the
approximation ratio deteriorate by a low order term only
(see, e.g., [3] for details).
Algorithm 4: MultilinearUSM(f, N )
1
2
3
4
5
6
x0 ← ∅, y0 ← N .
for i = 1 to n do
a′i ← F (xi−1 + {ui }) − F (xi−1 ).
b′i ← F (yi−1 − {ui }) − F (yi−1 ).
a′i ← max{ai , 0}, b′i ← max{bi , 0}.
a′i
*
xi ← xi−1 + a′ +b
′ · {ui } .
i
yi ← yi−1 −
7
8
i
b′i
a′i +b′i
· {ui }* .
return a random set R(xn ) (or equivalently R(yn )).
If a′i = b′i = 0, we
′
bi /(a′i + b′i ) = 0.
*
assume a′i /(a′i + b′i ) = 1 and
The main difference between Algorithms 2 and 4 is that
Algorithm 2 chooses each element with some probability,
whereas Algorithm 4 assigns a fractional value to the
element. This requires the following modifications to the
algorithm:
N
are replaced by the vectors
• The sets Xi , Yi ⊆ 2
N
xi , yi ∈ [0, 1] .
• Algorithm 4 uses the multilinear extension F instead
of the original submodular function f .
Theorem A.1. If one has an oracle access to F , Algorithm 4
is a (1/2)-approximation algorithm for NSM. Otherwise,
it can be implemented using sampling, and achieves an
approximation ratio of (1/2) − o(1).
The rest of this section is devoted to proving Theorem A.1. Similarly to Section III, define OP Ti , (OP T ∨
xi ) ∧ yi . Examine the sequence F (OP T0 ), . . . , F (OP Tn ).
Notice that OP T0 = OP T , i.e., the sequence starts with the
value of an optimal solution, and that OP Tn = xn = yn ,
i.e., the sequence ends at a fractional point whose value is
the expected value of the algorithm’s output. The following
lemma upper bounds the loss of two consecutive elements
in the sequence. Formally, F (OP Ti−1 )−F (OP Ti ) is upper
bounded by the average change in the value of the two
solutions maintained by the algorithm, i.e.,
1/2 · [F (xi ) − F (xi−1 ) + F (yi ) − F (yi−1 )].
Lemma A.2. For every 1 ≤ i ≤ n,
F (OP Ti−1 )−F (OP Ti ) ≤
1
· [F (xi ) − F (xi−1 ) + F (yi ) − F (yi−1 )] .
2
Before proving Lemma A.2, let us show that Theorem A.1
follows from it.
Proof of Theorem A.1: Summing up Lemma A.2 for
every 1 ≤ i ≤ n gives:
n
∑
[F (OP Ti−1 ) − F (OP Ti )] ≤
i=1
n
n
1 ∑
1 ∑
·
[F (xi ) − F (xi−1 )] + ·
[F (yi ) − F (yi−1 )] .
2 i=1
2 i=1
The above sum is telescopic. Collapsing it, we get:
F (OP T0 ) − F (OP Tn ) ≤
1
1
· [F (xn ) − F (x0 )] + · [F (yn ) − F (y0 )] ≤
2
2
F (xn ) + F (yn )
.
2
Recalling the definitions of OP T0 and OP Tn , we obtain
that F (xn ) = F (yn ) ≥ f (OP T )/2.
It is left to prove Lemma A.2.
Proof of Lemma A.2: By Lemma II.1, ai + bi ≥ 0,
therefore, it cannot be that both ai , bi are strictly less than
zero. Thus, we have 3 cases to consider.
Case 1 (ai ≥ 0 and bi ≤ 0): In this case a′i /(a′i +
′
bi ) = 1, and so yi = yi−1 , xi ← xi−1 ∨ {ui }. Hence,
F (yi ) − F (yi−1 ) = 0. Also, by our definition OP Ti =
(OP T ∨ xi ) ∧ yi = OP Ti−1 ∨ {ui }. Thus, we are left to
prove that:
F (OP Ti−1 )−F (OP Ti−1 ∨ {ui }) ≤
1
· [F (xi ) − F (xi−1 )] = ai /2 .
2
If ui ∈ OP T , then the left hand side of the above equation
is 0, which is clearly no larger than the non-negative ai /2.
If ui ∈
̸ OP T , then:
F (OP Ti−1 ) − F (OP Ti−1 ∨ {ui }) ≤
F (yi−1 − {ui }) − F (yi−1 ) = bi ≤ 0 ≤ ai /2 .
The first inequality follows from submodularity since
OP Ti−1 = ((OP T ∨ xi−1 ) ∧ yi−1 ) ≤ yi−1 − {ui } (note
that in this case (yi−1 )ui = 1 and (OP Ti−1 )ui = 0).
Case 2 (ai < 0 and bi ≥ 0): This case is analogous
to the previous one, and therefore, we omit its proof.
Case 3 (ai ≥ 0 and bi > 0): In this case a′i = ai , b′i =
i
i
bi and so, xi ← xi−1 + aia+b
· {ui } and yi ← yi−1 − aib+b
·
i
i
{ui }. Therefore, we have,
F (xi ) − F (xi−1 ) =
a2i
ai
· [F (xi−1 ∨ {ui }) − F (xi−1 )] =
.
ai + bi
ai + bi
(4)
A similar argument shows that:
F (yi ) − F (yi−1 ) =
b2i
.
ai + bi
(5)
Next, we upper bound F (OP Ti−1 ) − F (OP Ti ). For simplicity, let us assume ui ̸∈ OP T (the proof for the other
case is similar). Recall, that OP Ti = (OP T ∨ xi ) ∧ yi .
F (OP Ti−1 ) − F (OP Ti ) =
ai
· [F (OP Ti−1 ) − F (OP Ti−1 ∨ {ui })] ≤
ai + bi
ai bi
ai
· [F (yi−1 − {ui }) − F (yi−1 )] =
.
ai + bi
ai + bi
(6)
The inequality follows from the submodularity of f since,
OP Ti−1 = ((OP T ∨ xi−1 ) ∧ yi−1 ) ≤ yi−1 − {ui }
(note again that in this case (yi−1 )ui = 1 and
(OP Ti−1 )ui = 0). Plugging (4), (5) and (6) into the
inequality that we need to prove, we get the following:
ai bi
1 a2 + b2i
≤ · i
,
ai + bi
2 ai + bi
which can be easily verified.
B. Proof of Theorem I.3
In this section we prove that Algorithm 3 is a randomized
linear time (3/4)-approximation algorithm for SSAT. As a
first step we state the following useful lemma.
Lemma A.3. For every 1 ≤ i ≤ n,
E[g(OP Ti−1 ) − g(OP Ti )] ≤
1
· E [(g(Xi ) − g(Xi−1 ) + g(Yi ) − g(Yi−1 )] ,
(7)
2
where expectations are taken over the random choices of the
algorithm.
The proof of this lemma is deferred to a full version of
the paper. Let us just note that the proof follows the lines of
Lemma III.1. Let us show that the approximation guarantee
of Theorem I.3 follows from the above lemma. The proof
that Algorithm 3 can be implemented to run in linear time
is deferred to a full version of the paper.
Proof of approximation guarantee of Algorithm 3:
Summing up Lemma A.3 for every 1 ≤ i ≤ n we get,
n
∑
E[g(OP Ti−1 ) − g(OP Ti )] ≤
i=1
n
1 ∑
·
E[g(Xi ) − g(Xi−1 ) + g(Yi ) − g(Yi−1 )] .
2 i=1
The above sum is telescopic. Collapsing it, we get:
E[g(OP T0 ) − g(OP Tn )]
1
≤ · E[g(Xn ) − g(X0 ) + g(Yn ) − g(Y0 )]
2
E[g(Xn ) + g(Yn ) − g(Y0 )]
≤
.
2
Recalling the definitions of OP T0 and OP Tn , we obtain
that:
E[g(Xn )] = E[g(Yn )] ≥ g(OP T )/2 + g(Y0 )/4.
The approximation ratio now follows from the observation
that Y0 satisfies all clauses of Ψ, and therefore, g(Y0 ) ≥
g(OP T ).